Prime Numbers

9.7 Exercises 531
adjustment step. Study the phenomenon raised in the text after the algorithm,
namely, that of reduced error in the balanced option. There exist some
numerical studies of this, together with some theoretical conjectures (see
[Crandall and Fagin 1994], [Crandall et al. 1999] and references therein), but
very little is known in the way of error bounds that are both rigorous and
pragmatic.
9.56. Show that if p =2 q − 1withq odd and x ∈{0,...,p − 1}, then
x 2 mod p can be calculated using two size-(q/2) multiplies. Hint: Represent
x = a + b2 (q+1)/2 and relate the result of squaring x to the numbers
(a + b)(a +2b) and(a − b)(a − 2b).
This interesting procedure gives nothing really new—because we already know
that squaring (in the grammar-school range) is about half as complex as
multiplication—but the method here is a different way to get the speed
doubling, and furthermore does not involve microscopic intervention into the
squaring loops as discussed for equation (9.3).
9.57. Do there always exist primes p1,...,pr required in Algorithm 9.5.20,
and how does one find them?
9.58. Prove, as suggested by the statement of Algorithm 9.5.20, that any
convolution element of x × y in that algorithm is indeed bounded by NM 2 .
For application to large-integer multiplication, can one invoke balanced
representation ideas, that is, considering any integer (mod p) as lying in
[−(p +1)/2, (p − 1)/2], to lower the bounding requirements, hence possibly
reducing the set of CRT primes?
9.59. For the discrete, prime-based transform (9.33) in cases where g has a
square root, h 2 = g, answer precisely: What is a closed form for the transform
element Xk if the input signal is defined x = h j2 , j = 0,...,p − 1?
Noting the peculiar simplicity of the Xk, find an analogous signal x having
N elements in the complex domain, for which the usual, complex-valued FFT
has a convenient property for the magnitudes |Xk|. (Such a signal is called
a “chirp” signal and has high value in testing FFT routines, which must, of
course, exhibit a numerical manifestation of the special magnitude property.)
9.60. For the Mersenne prime p =2 127 − 1, exhibit an explicit primitive
64-th root of unity a + bi in F ∗ p 2.
9.61. Show that if a + bi is a primitive root of maximum order p 2 − 1inF ∗ p 2
(with p ≡ 3 (mod 4), so that “i” exists), then a 2 +b 2 must be a primitive root
of maximum order p − 1inF ∗ p. Is the converse true?
Give some Mersenne primes p =2 q − 1 for which 6 + i is a primitive root
in F ∗ p 2.
9.62. Prove that the DGT integer convolution Algorithm 9.5.22 works.